This webpage provides supporting information for the manuscript “Phylogenetic generalised linear mixed-effects modelling with glmmTMB R package”. The webpage is organised into three sections: (1) First, we describe the simulated models and data-generating mechanisms for each of the five evaluated R packages for fitting PGLMMs (2) We then use the simulated data to illustrate compilation and sampling runtimes for the Bayesian MCMC models. (3) Finally, we presents two case studies illustrating the application of these methods to ecological and evolutionary data, the first on bird color traits and the second on plant traits.
We demonstrate and explain how to fit phylogenetic generalized mixed models (PGLMM) using five different packages which we used in our simulation study: glmmTMB, phyr, MCMCglmm, brms, and INLA. The models are fitted to one simulated dataset using a randomly generated phylogenetic tree and a set of model parameters. We compare the performance of these packages in terms of runtime, accuracy, and bias of the fixed effect mean and random effect variance estimates.
We assume a set of trait measures \(y_{ij}\) repeated measure (observation) \(i\) and for each species \(j\). The model is specified as follows:
\(p_i \sim \mathcal{N}(0, \sigma^2_{\text{p}} \mathbf{A})\): species-level effect (phylogenetic), where \(\mathbf{A}\) is the phylogenetic correlation matrix
First we specify the parameter values for one simulation run. We assume the following:
Code
### set up model parameters seed <-1b0 <-1# fixed effect interceptb1 <-1.5# fixed effect slopek.species <-30# number of speciesn.reps <-10# number of repeated measures per species (assuming a balanced design)sigma2.n <-0.25# variance of non-phylogenetic effectsigma2.p <-0.25# variance of phylogenetic effectsigma2.e <-0.2# residual error variance
Simulate data based on these parameter values:
Code
set.seed(seed)# set up k.obs (number of observations per species)k.obs <- n.reps # assumes a balanced design# species id sp.id <-rep(seq_len(k.species), times=k.obs)# total number of observationsn <-length(sp.id)# simulate simple dataframe with covariate (x) variablex <-runif(n, 10, 20) dat <-data.frame(obs =1:n, x = x, species = sp.id)# simulate tree and obtain phylo matrixtree <-rtree(k.species, tip.label =seq_len(k.species))tree <-compute.brlen(tree, power=1) # power of 1 means ultra-metric treephylo.mat <-vcv(tree, corr =TRUE) ## we want a correlation matrix (bounded by -1 and 1)phylo.mat <- phylo.mat[order(as.numeric(rownames(phylo.mat))), order(as.numeric(rownames(phylo.mat)))]# Simulate response variable (phen) based on cofactor and phylogenetic matrixu.s <-rnorm(k.species, 0, sqrt(sigma2.n))[sp.id]u.p <-mvrnorm(1, mu=rep(0, k.species), Sigma=sigma2.p*phylo.mat)[sp.id]ei <-rnorm(n, 0, sqrt(sigma2.e))# get estimates of yyi <- b0 + b1*x + u.s + u.p + ei### append all to dataframedat <-cbind(dat, u.s, u.p, ei, b0, b1, yi)dat$phylo <- dat$species # phylo ID variable (same as species) - needs to be numeric to work with INLAdat$species <-factor(dat$species) # format species variable for modelsdat$sp <- dat$species # create sp variable (for phyr)dat$g <-1# add variable g constant (for glmmTMB)
To fit a PGLMM using glmmTMB package we specify the species-level random effect with phylogenetic relatedness using the propto covariance structure. In the first part we specify the random intercept (0 + species), followed by the grouping variable g (which we assume here as constant), and then the phylogenetic correlation matrix phylo.mat.
The model output will provide the fixed effect estimates, random effect variance estimates, and the residual variance estimate. The random effect variance estimates will be on the standard deviation scale, so we square them to get the variance estimates.
Code
# repeated measures per speciestime.glmmTMB <-system.time({ model_glmmTMB <-glmmTMB(yi ~ x + (1|species) +propto(0+ species|g, phylo.mat),data = dat,REML =TRUE)})
Check if the model converged (i.e. returns TRUE if the Hessian is positive definite, for more details read here: https://cran.r-project.org/web/packages/glmmTMB/vignettes/troubleshooting.html):
Code
model_glmmTMB$sdr$pdHess
[1] TRUE
Run model with phyr
To fit the model using phyr package we use the cov_ranef argument to specify the phylogenetic tree directly. The random effect term (1|sp__) indicates that we want to include both phylogenetic and non-phylogenetic random effects. The underscores __ indicate that we want to include phylogenetic correlations in the model.
Run model with MCMCglmm
To fit the MCMCglmm model we first we need to set up a precision matrix for the phylogenetic random effect which is required by MCMCglmm. We use the inverseA function from the ape package to obtain the inverse of the phylogenetic covariance matrix. The prior list specifies the priors for the random effects and residual variance. The ginverse argument is used to specify the inverse of the phylogenetic covariance matrix. Then we specify the number of iterations, burn-in, and thinning parameters for the MCMC sampling. Here we increased the number of iterations by 30 times the default value to ensure convergence and stability of the MCMC chains, but this could be further increased. Usually you would want to set these to some reasonably large values.
Next we check the effective sample size (ESS) and the Heidelberger diagnostic test which checks if the MCMC chains have converged and are stationary. The ESS should be above 400 for both fixed and random effects (Vehtari et al., 2021), and the Heidelberger diagnostic should return TRUE for all parameters.
Stationarity start p-value
test iteration
(Intercept) passed 1 0.184
x passed 1 0.580
species passed 1 0.767
phylo passed 1 0.855
units passed 1 0.490
Halfwidth Mean Halfwidth
test
(Intercept) passed 0.845 0.005440
x passed 1.510 0.000114
species passed 0.149 0.002052
phylo passed 0.660 0.017493
units passed 0.208 0.000203
Run model with brms
To fit the model using the brms package we specify the random effects using the (1|species) term for non-phylogenetic random effects and gr(phylo, cov = phylo.mat) for phylogenetic random effects. The data2 argument is used to pass the phylogenetic correlation matrix to the model (the same matrix format as for glmmTMB). We set the number of iterations, chains, and cores to reasonable values to ensure convergence and stability of the MCMC chains. Here we increased the number of iterations by 10 times the default value to ensure convergence and stability of the MCMC chains, but this could be further increased (and should be if ESS are low and Rhat values are higher than 1.01).
Check that the maximum effective sample size (ESS) of the model is high enough and check the Rhat value is below 1.01 (Vehtari et al. 2021):
# check RHat value is below 1.01 (Vehtari et al. 2021)max(rhat(model_brms))<1.01
[1] TRUE
Code
# alternatively, we can use the bayestestR package to check the diagnostics of fixed effectsprint(bayestestR::diagnostic_posterior(model_brms), digits =4)
For the INLA package we set up the model using the f() function to specify the random effects. The model = "iid" argument indicates that we want to use an independent and identically distributed (iid) random effect for species, while the model = "generic0" argument is used for the phylogenetic random effect. The Cmatrix argument is used to specify the phylogenetic correlation matrix. We also set up a penalizing complexity prior for the precision of the phylogenetic random effect (suggested by xxxx don’t have a ref but can mention it?).
Code
# set up recommended penalizing complexity priors pcprior =list(prec =list(prior="pc.prec", param =c(20, 0.1)))time.inla <-system.time({ model_inla <-inla(yi ~ x +f(species, model ="iid") +f(phylo, ## this needs to be a numeric to workmodel ="generic0",Cmatrix = phylo.prec.mat,hyper=pcprior),family ="gaussian",data = dat)})fit_inla <-summary(model_inla)
Extract model estimates
First we extract the covariate \(x\) estimate and confidence interval for each model:
Code
## Format run time for each model ---------time.phyr <-as.numeric(time.phyr[3])time.glmmTMB <-as.numeric(time.glmmTMB[3])time.brms <-as.numeric(time.brms[3])time.mcmc <-as.numeric(time.mcmc[3])time.inla <-as.numeric(time.inla[3])## Fixed effect estimates -------# phyrcoefs_phyr <-as.data.frame(fixef(model_phyr))coefs_phyr$conf.low[2] <- coefs_phyr$Value[2] - coefs_phyr$Std.Error[2]*1.96coefs_phyr$conf.high[2] <- coefs_phyr$Value[2] + coefs_phyr$Std.Error[2]*1.96# glmmTMBcoefs_tmb <-as.data.frame(confint(model_glmmTMB, parm="beta_"))# brmscoefs_brm <-as.data.frame(tidy(model_brms, effects="fixed", conf.int=TRUE))# MCMCglmm coefs_mcmc <-as.data.frame(tidy(model_mcmc, effects="fixed", conf.int=TRUE))# INLAcoefs_inla <-as.data.frame(fit_inla$fixed)## Combine fixed effect results -----------------res_fixed <-data.frame(model =c("phyr", "glmmTMB", "brms", "MCMCglmm", "INLA"),species_size = k.species,sample_size = n,run_time =c(time.phyr, time.glmmTMB, time.brms, time.mcmc, time.inla),b0 =rep(dat$b0[1], 5),b1 =rep(dat$b1[1], 5),mu =c(coefs_phyr$Value[2], coefs_tmb$Estimate[2], coefs_brm$estimate[2], coefs_mcmc$estimate[2], coefs_inla$mean[2]),mu_ci_low =c(coefs_phyr$conf.low[2], coefs_tmb$`2.5 %`[2], coefs_brm$conf.low[2], coefs_mcmc$conf.low[2], coefs_inla$`0.025quant`[2]),mu_ci_high =c(coefs_phyr$conf.high[2], coefs_tmb$`97.5 %`[2], coefs_brm$conf.high[2], coefs_mcmc$conf.high[2], coefs_inla$`0.975quant`[2]),stringsAsFactors =FALSE)## Show table of results of runtime and fixed effects ------kable(res_fixed, caption ="Runtime and fixed effect estimates of the simulated model",col.names =c("Model", "Species size", "Sample size", "Run time (s)", "b0", "b1", "Estimate (b1)", "CI low (b1)", "CI high (b1)"),digits =3,format ="html")
Runtime and fixed effect estimates of the simulated model
Model
Species size
Sample size
Run time (s)
b0
b1
Estimate (b1)
CI low (b1)
CI high (b1)
phyr
30
300
0.55
1
1.5
1.51
1.491
1.530
glmmTMB
30
300
1.00
1
1.5
1.51
1.491
1.530
brms
30
300
489.67
1
1.5
1.51
1.490
1.530
MCMCglmm
30
300
134.45
1
1.5
1.51
1.490
1.530
INLA
30
300
5.64
1
1.5
1.51
1.490
1.529
Extract the variance component estimates for each model and compare:
Code
#### Random effect estimates -------# get phyr random effect variance estimates var_re_phyr <-c(as.numeric(model_phyr$ss[2])^2, #phylogenetic as.numeric(model_phyr$ss[1])^2, #non-phylogenetic as.numeric(model_phyr$ss[3])^2) #residual # combine into dataframesigma2_phyr <-data.frame(model ="phyr",group =c("phylo", "species", "Residual"),term ="var",estimate = var_re_phyr,std.error =NA,conf.low =NA, conf.high =NA)# get glmmTMB random effect variance estimates (by default it is on the standard deviation scale)re_tmb <-as.data.frame(confint(model_glmmTMB, parm="theta_"))species_tmb <- re_tmb[1, ]phylo_tmb <- re_tmb[2, ]# combine into dataframesigma2_tmb <-data.frame(model ="glmmTMB",group =c("phylo", "species", "Residual"),term ="var",estimate =c(phylo_tmb$Estimate^2, #phylo variance estimates species_tmb$Estimate^2, #non-phylo variance estimatessigma(model_glmmTMB)^2),std.error =NA, #conf.low =c(phylo_tmb$`2.5 %`, species_tmb$`2.5 %`, NA), # Replace with the residual var CI if availableconf.high =c(phylo_tmb$`97.5 %`, species_tmb$`97.5 %`, NA) # Replace with the residual var CI if available)# Compute variance, SE (delta method), and CI on variance scale)var_est <- re_tmb$Estimate^2var_se <-2* re_tmb$Estimate * (re_tmb$`97.5 %`- re_tmb$`2.5 %`) / (2*1.96)var_ci_low <- re_tmb$`2.5 %`^2var_ci_high <- re_tmb$`97.5 %`^2# Residual varianceresid_var <-sigma(model_glmmTMB)^2# Combine resultssigma2_tmb <-data.frame(model ="glmmTMB",group =c("phylo", "species", "Residual"),term ="var",estimate =c(var_est, resid_var),std.error =c(var_se, NA),conf.low =c(var_ci_low, NA),conf.high =c(var_ci_high, NA))# get brms random effect variance estimates (standard deviation scale)sigma_brms <-tidy(model_brms, effects="ran_pars")sigma2_brms <- sigma_brms %>%mutate(model="brms",term=str_replace(term, "sd", "var"),estimate=estimate^2) %>%##compute variance estimates dplyr::select(model, group, term, estimate, std.error, conf.low, conf.high)# get MCMCglmm random effect estimates (variance scale)sigma2_mcmc <-tidy(model_mcmc, effects="ran_pars", conf.int=TRUE)sigma2_mcmc <- sigma2_mcmc %>%mutate(model="MCMCglmm",group=str_replace(group,"animal", "phylo")) %>% dplyr::select(model, group, term, estimate, std.error, conf.low, conf.high)# get INLA random effect estimates (precision scale i.e. inverse variance)re_inla <-1/fit_inla$hyperparsigma2_inla <-data.frame(model ="INLA",group =c( "Residual", "species", "phylo"),term ="var",estimate = re_inla$mean,std.error = fit_inla$hyperpar$sd,conf.low = re_inla$`0.025quant`,conf.high = re_inla$`0.975quant`)# merge fixed results togethers2 <-as.data.frame(rbind(sigma2_phyr, sigma2_tmb, sigma2_brms, sigma2_mcmc, sigma2_inla))# get subsets for each groups2_phylo <- s2 %>%filter(group=="phylo")s2_sp <- s2 %>%filter(group=="species")s2_res <- s2 %>%filter(group=="Residual")## Combine random effect var estimates results -----------------res_rand <-data.frame(model =c("phyr", "glmmTMB", "brms", "MCMCglmm", "INLA"),species_size = k.species,sample_size = n,run_time =c(time.phyr, time.glmmTMB, time.brms, time.mcmc, time.inla),sigma2_phylo = s2_phylo$estimate,sigma2_species = s2_sp$estimate,sigma2_residual = s2_res$estimate,stringsAsFactors =FALSE)## Show table of results of runtime and random variance ------kable(res_rand, caption ="Runtime and random component variance estimates of the simulated model",col.names =c("Model", "Species size", "Sample size", "Run time (s)", "Phylo variance est.", "Non-phylo variance est.", "Residual variance est."),digits =3,format ="html")
Runtime and random component variance estimates of the simulated model
Model
Species size
Sample size
Run time (s)
Phylo variance est.
Non-phylo variance est.
Residual variance est.
phyr
30
300
0.55
0.346
0.639
0.206
glmmTMB
30
300
1.00
0.132
0.427
0.206
brms
30
300
489.67
0.568
0.140
0.208
MCMCglmm
30
300
134.45
0.660
0.149
0.208
INLA
30
300
5.64
0.549
0.077
0.205
3 Bayesian MCMC model runtime
Here we provide a breakdown of the compilation and sampling runtimes for the Bayesian MCMC models using the above simulated repeated measures dataset. The runtimes are provided for each model fitted using the MCMCglmm and brms packages. Note: the compilation time is the time taken to compile the model, while the sampling time is the time taken to sample from the posterior distribution of the model parameters.
First we set up the MCMCglmm model.
MCMCglmm
Set up MCMCglmm tuning parameters - usually set to some reasonably large values through trial and error.
SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 0.000219 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 2.19 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: There aren't enough warmup iterations to fit the
Chain 1: three stages of adaptation as currently configured.
Chain 1: Reducing each adaptation stage to 15%/75%/10% of
Chain 1: the given number of warmup iterations:
Chain 1: init_buffer = 3
Chain 1: adapt_window = 19
Chain 1: term_buffer = 2
Chain 1:
Chain 1: Iteration: 1 / 26925 [ 0%] (Warmup)
Chain 1: Iteration: 25 / 26925 [ 0%] (Sampling)
Chain 1: Iteration: 2716 / 26925 [ 10%] (Sampling)
Chain 1: Iteration: 5408 / 26925 [ 20%] (Sampling)
Chain 1: Iteration: 8100 / 26925 [ 30%] (Sampling)
Chain 1: Iteration: 10792 / 26925 [ 40%] (Sampling)
Chain 1: Iteration: 13484 / 26925 [ 50%] (Sampling)
Chain 1: Iteration: 16176 / 26925 [ 60%] (Sampling)
Chain 1: Iteration: 18868 / 26925 [ 70%] (Sampling)
Chain 1: Iteration: 21560 / 26925 [ 80%] (Sampling)
Chain 1: Iteration: 24252 / 26925 [ 90%] (Sampling)
Chain 1: Iteration: 26925 / 26925 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0.036 seconds (Warm-up)
Chain 1: 3.657 seconds (Sampling)
Chain 1: 3.693 seconds (Total)
Chain 1:
SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 0.000302 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 3.02 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: There aren't enough warmup iterations to fit the
Chain 1: three stages of adaptation as currently configured.
Chain 1: Reducing each adaptation stage to 15%/75%/10% of
Chain 1: the given number of warmup iterations:
Chain 1: init_buffer = 3
Chain 1: adapt_window = 19
Chain 1: term_buffer = 2
Chain 1:
Chain 1: Iteration: 1 / 26925 [ 0%] (Warmup)
Chain 1: Iteration: 25 / 26925 [ 0%] (Sampling)
Chain 1: Iteration: 2716 / 26925 [ 10%] (Sampling)
Chain 1: Iteration: 5408 / 26925 [ 20%] (Sampling)
Chain 1: Iteration: 8100 / 26925 [ 30%] (Sampling)
Chain 1: Iteration: 10792 / 26925 [ 40%] (Sampling)
Chain 1: Iteration: 13484 / 26925 [ 50%] (Sampling)
Chain 1: Iteration: 16176 / 26925 [ 60%] (Sampling)
Chain 1: Iteration: 18868 / 26925 [ 70%] (Sampling)
Chain 1: Iteration: 21560 / 26925 [ 80%] (Sampling)
Chain 1: Iteration: 24252 / 26925 [ 90%] (Sampling)
Chain 1: Iteration: 26925 / 26925 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0.225 seconds (Warm-up)
Chain 1: 631.601 seconds (Sampling)
Chain 1: 631.826 seconds (Total)
Chain 1:
Code
runtime_2_run$elapsed - runtime_2_pre$elapsed
[1] 624.9
4 Case studies
4.1 Background
The following case studies illustrate how phylogenetic mixed models can be applied to diverse questions in ecology and evolution. The first examines patterns of plumage colour in birds, while the second explores macroevolutionary patterns in plant traits. Together, they demonstrate the flexibility of these methods across taxa and research questions.
All models, results, and datasets presented in these case studies are intended solely for illustrative purposes. They are designed to demonstrate analytical approaches rather than to provide definitive biological insights. No substantive conclusions should be drawn from these analyses.
4.2 Case study 1: Bird color evolution
Birds are a valuable system for studying ecological and evolutionary processes because their diversity in form, behaviour, and coloration is well documented across many species and habitats. Variation in plumage colour, in particular, can provide insights into mechanisms such as sexual selection, camouflage, and signalling. By examining these traits, researchers can better understand how environmental and evolutionary pressures shape biodiversity.
Data overview
Load data bird spectrum data:
Code
# Load bird spectrum data bird_data <-read_delim("data/Spec_IndivReg_Coralie.csv", delim =";", escape_double =FALSE, locale =locale(decimal_mark =","), trim_ws =TRUE)# summary of birds species and genuslength(table(bird_data$genus_original)) # 446 genus
[1] 446
Code
length(table(bird_data$sci_name_Jetz)) # 949 species
[1] 949
Code
#length(table(bird_data$sci_name_original)) # 952 original species# Remove the specified species from bird_databird_data <- bird_data |>filter(!sci_name_Jetz %in%c("Basileuterus_rufifrons", "Malurus_lamberti", "Malurus_splendens"))########## Notes ########## missing values ---> not sure why?#na_count <- colSums(is.na(dat2))#na_count[na_count > 0]# individuals a-c are male and d-f are female#table(bird_data$sex, bird_data$individual)# number of measurements?#table(bird_data$Nmeasured)# duplicates per individual per body region#dup <- dat |># summarise(n = n(), # .by = c(wl, individual_nonrep, sex, body_region))#r <- dup[which(dup$n>1),]#table(r$individual_nonrep)
Some notes:
The dataset contains data for 949 bird species, with 446 unique genera.
Spectral values are normalised reflectance data.
Nmeasured is the number of measurements per body patch.
Each body region measured 5 times (then averaged).
These species seem to have two measurements per body region per individual, which we remove to facilitate the analysis.:
Now we want to create a wavelength dataset using the pavo package given the spectral reflectance data.
Code
# set up wavelength dataset (wavelengths columns 300 to 474)dat <- bird_data |>pivot_longer(cols =`300`:`700`, names_to="wl", values_to="refl") |> dplyr::select(individual_nonrep, body_region, sex, wl, refl) |>mutate(wl =as.numeric(wl))# pivot so each column is an individual body region measurementdat2 <- dat |>pivot_wider(names_from =c("individual_nonrep", "sex", "body_region"),values_from ="refl", names_sep =".")# create spectral dataset with pavospecs <-as.rspec(dat2)# some examples of individual birds body region color spectrumplot(Acrocephalus_palustris_a.Male.throat ~ wl, type="l", data=specs, title="Acrocephalus palustris (throat)")
Code
plot(Alle_alle_a.Male.throat ~ wl, type="l", data=specs, title="Alle alle (throat)")
Code
plot(Aplonis_metallica_a.Male.wing_cov ~ wl, type ="l", data = specs, title="Aplonis metallica (wing covert)")
Code
# use procspec to adjust negative values by shifting them by 10# (this maybe due to darker colors)specs <-procspec(specs, fixneg="addmin")plot(specs, select =10)
Obtain spectral shape descriptors
Using the wavelength dataset we can now obtain spectral shape descriptors. These descriptors will be used in the model. Here we will focus on four descriptors of interest: brightness (B1), spectral slope (S1), spectral curvature (S9), and hue (H4). More information about the spectral shape descriptors can be found here: https://book.colrverse.com/spectral-shape-descriptors.html
Code
# Obtain spectral shape descriptorsspec.des <-summary(specs, subset =c("B1", "B2", "S9", "H4")) ## using subset makes it faster to run by selecting the shapes of interestdev.off()
null device
1
Code
# distribution of B1ggplot(spec.des, aes(x = B1)) +geom_histogram(bins =30, colour ="black", fill ="grey") +labs(x ="B1", title ="B1: Total brightness") +theme_bw()# distribution of S9ggplot(spec.des, aes(x = S9)) +geom_histogram(bins =30, colour ="black", fill ="grey") +labs(x ="S9", title ="S9: Carotenoid chroma") +theme_bw()# vdistribution o fH4ggplot(spec.des, aes(x = H4)) +geom_histogram(bins =30, colour ="black", fill ="grey") +labs(x ="H4", title ="H4: Hue (segment classification)") +theme_bw()
The first dataset is the shape descriptors dataset, which we will use to model the continuous trait (brightness).
Obtain carotenoid datasets: binary and ordinal traits
We will use the carotenoid dataset to model the presence/absence of carotenoid coloration in birds. This dataset is based on the spectral reflectance data and the spectral shape descriptors.
Let’s set up the dataset for modelling a binary trait (absence/presence of carotenoid color):
Code
# 1) get human visual model (CIE 10 degree observed under D65 "day light")vm <-vismodel(specs, visual ="cie10", illum ="D65", bkg ="ideal", relative =FALSE)# 2) convert to CIELAB colorspace (to get hue and chroma)lab <-colspace(vm, space ="cielab") # 3) get chroma and hue angle (converting CIELAB to CIELCh)L <- lab$La <- lab$ab <- lab$bC <-sqrt(a^2+ b^2)h <- (atan2(b, a) *180/ pi) %%360# degrees in 0 to 360# 4) thresholds for carotenoid range and saturationh_lower <-330# start of redh_upper <-100# end of yellowC_min <-15# min chromaL_min <-20# avoid very dark samples# 5) Hue range test in_range <- h >= h_lower | h <= h_upper# 6) COmpute binary trait: 1 = carotenoid like colour present, 0 = absentpresent <-as.integer(in_range & C >= C_min & L >= L_min)# set up datasetcarot.dat.all <-data.frame(rowname =rownames(lab),L = L, a = a, b = b, C = C, h = h,carotenoid = present)# merge with spec.data based on "rowname" columncarot.dat.all <-merge(spec.dat, carot.dat.all, by.x ="rowname", by.y ="rowname", all =TRUE)# create individual id variablecarot.dat.all$indiv_rep <-sub(".*_([a-f])\\..*$", "\\1", carot.dat.all$rowname)# number of observations with carotenoid like colourtable(carot.dat.all$carotenoid)
carot.dat <- carot.dat.all# save as csvwrite.csv(carot.dat, file="data/carotenoid_data_for_model.csv", row.names =FALSE)
Now let’s obtain the dataset where we summarise for each individual bird the proportion of body region with carotenoid color presence (ordinal data trait), and save it for modelling later on:
Code
carot.ordinal <- carot.dat.all |>group_by(species, sex, indiv_rep) |>summarise(prop_carotenoid =mean(carotenoid), .groups ="drop")# save as csvwrite.csv(carot.ordinal, file="data/ordinal_data_for_model.csv", row.names =FALSE)
Phylogenetic correlation matrix set-up
Load and view phylogenetic tree:
Code
# Load bird tree (consensus tree = "combined tree")bird.tree <-read.tree("data/Stage2_Hackett_MCC_no_neg.tre")### Prune bird treebird.pruned <-keep.tip(bird.tree, bird_data$sci_name_Jetz)# check whether names match in data and treecheck <-name.check(bird.pruned, bird_data$sci_name_Jetz, sort(bird.pruned$tip.label))# plot treeplotTree(bird.pruned, ftype="i", fsize=0.4, lwd=1, type="fan")
Code
dev.off()
null device
1
Set up correlation matrix for glmmTMB model and check it corresponds to the species labels in the data:
Code
# set up phylogenetic correlation matrixphylo.mat <-vcv(bird.pruned, corr =TRUE) phylo.mat <- phylo.mat[sort(rownames(phylo.mat)), sort(rownames(phylo.mat))]saveRDS(phylo.mat, file ="data/phylo_matrix.rds")# checks # length(colnames(phylo.mat))==length(table(spec.dat$species))# all(head(rownames(phylo.mat))==head(colnames(phylo.mat)))# head(table(spec.dat$species))
Models
Continuous trait (brightness): we compare models with distributions of gaussian, lognormal, gamma, and skew normal.
Binary trait (absence/presence of color): binomial distribution.
Ordinal trait (100% of body region with color): beta binomial.
1. Continuous trait
Code
# load dataspec.dat <-read_csv("data/spec_data_for_model.csv")phylo.mat <-readRDS("data/phylo_matrix.rds")# load library library(glmmTMB)# add grouping variable (set it to 1) - this is necessary to fit the glmmTMB modelspec.dat$g <-1
Modelling total brightness (B1)
The brightness trait is right-skewed (as shown above), which is consistent with multiplicative evolutionary change. To identify an appropriate sampling distribution, we will fit four models with an identical linear predictor and random effects structure:
Gaussian to model \(\log(B1)\)
Gamma with a log link to model \(B1\)
For each model we will examine simulated standardised residuals using to assess nonlinearity and heteroscedasticity. If two or more models perform similarly, we will prefer the model with clearer interpretation on the original scale.
Code
# # Little checks ---------# library(fitdistrplus)# descdist(spec.dat$B1, discrete=FALSE)# gamma_dist <- fitdist(spec.dat$B1, "gamma")# plot(gamma_dist)# norm_dist <- fitdist(log(spec.dat$B1), "norm")# plot(norm_dist)# lnorm_dist <- fitdist(spec.dat$B1, "lnorm")# plot(lnorm_dist)# Fit models --------------# normaltime_norm <-system.time( m1 <-glmmTMB(log(B1) ~ body_region * sex + (1|species) +propto(0+ species|g,phylo.mat),family =gaussian(),data = spec.dat) # don't use REML (to get AIC) )# Gamma distributiontime_gamma <-system.time( m2 <-glmmTMB(B1 ~ body_region * sex + (1|species) +propto(0+ species|g,phylo.mat),family =Gamma(link ="log"),data = spec.dat))# # Lognormal ---> not working get an error that the link function is not supported?# t_lnorm <- system.time(# modb1_lnorm <- glmmTMB(B1 ~ body_region * sex + (1|species) + propto(0 + species|g,phylo.mat), family = lognormal(link = "log"), data = spec.dat)# )# Get model info ---------------------------# check whether model has postive definite hessianb1_output <-data.frame(model =c("Gaussian (log)", "Gamma"),convergence =c(m1$sdr$pdHess, m2$sdr$pdHess),runtime =c(time_norm[["elapsed"]], time_gamma[["elapsed"]]))b1_output
On the log mean scale of the Gamma model, the phylogenetic species effect explained 69.8% of the total species level random effect variance i.e. this represents the degree of phylogenetic signal in the overall variance sourced from species.
To assess differences in total brightness between sexes for each body region, we compute marginal means from the fitted model and perform pairwise comparisons between females and males. The resulting ratios (female/male) and their confidence intervals are then plotted to visualise the magnitude and direction of differences across body regions.
Code
emm_b1 <-emmeans(m2, ~ sex | body_region, type ="response")b1_sex_diff <-contrast(emm_b1, method ="pairwise")## Transform to response scale as a ratiosex_diff_df <-as.data.frame(b1_sex_diff)sex_diff_df$lower.CL <- sex_diff_df$ratio - sex_diff_df$SEsex_diff_df$upper.CL <- sex_diff_df$ratio + sex_diff_df$SE## plot pairwise differences between female and male total brightnesssex_diff_df |>arrange(ratio) |>mutate(body_region =factor(body_region, unique(body_region), ordered = T)) |>ggplot(aes(y = body_region, x = ratio, color = body_region)) +geom_point(size =3) +geom_errorbar(aes(xmin = lower.CL, xmax = upper.CL), width =0.2) +geom_vline(xintercept =1, linetype ="dashed", colour ="grey40") +labs(y ="Body Region", x ="Female / Male Ratio", title ="Brightness") +theme_bw() +theme(legend.position ="none") # Optional: remove legend if redundant
Ratios \(>1\) indicate greater brightness in females, while ratios \(<1\) indicate greater brightness in males.
2. Binary trait
Binary pigment traits, such as the presence of carotenoid colors, may be associated with distinct evolutionary and ecological drivers.
Here we want to model the absence or presence of carotenoid across all body regions accounting for sex and including species- and phylogeny-level random effects. We will use the carotenoid dataset to model based on the spectral reflectance data which we derived above.
Modelling carotenoid presence
We will fit two models: - Binomial model with a logit link function - beta binomial model with a logit link function
Code
# Fit models --------------# load datcarot.dat <-read_csv("data/carotenoid_data_for_model.csv")carot.dat$g <-1# binomial modeltime_binom <-system.time( m3 <-glmmTMB(carotenoid ~ body_region * sex + (1|species) +propto(0+ species|g,phylo.mat),family =binomial(link ="logit"),data = carot.dat))# beta binomialtime_bbinom <-system.time( m4 <-glmmTMB(carotenoid ~ body_region * sex + (1|species) +propto(0+ species|g,phylo.mat),family =betabinomial(link ="logit"),data = carot.dat))# Get model info -----------carotmod_output <-data.frame(model =c("Binomial", "Beta binomial"),convergence =c(m3$sdr$pdHess, m4$sdr$pdHess),runtime =c(time_binom[["elapsed"]], time_bbinom[["elapsed"]]),AIC =c(AIC(m3), AIC(m4)))carotmod_output
We found that the binomial model has improved model fit in AIC and in the residual plots.
Model estimates
Get model estimates:
Code
library(emmeans)# Estimated marginal means on the response scale (odds)emm_m3 <-emmeans(m3, ~ sex | body_region, type ="response")# get contrasts (odds ratios)m3_sex_diff <-contrast(emm_m3, method ="pairwise") |>summary(infer =TRUE, type ="response")# Put into data frame for plottingsex_diff_df <-as.data.frame(m3_sex_diff) |>rename(ratio = odds.ratio, lower.CL = asymp.LCL, upper.CL = asymp.UCL)# Plotsex_diff_df |>arrange(ratio) |>mutate(body_region =factor(body_region, unique(body_region), ordered =TRUE)) |>ggplot(aes(y = body_region, x = ratio, colour = body_region)) +geom_point(size =3) +geom_errorbar(aes(xmin = lower.CL, xmax = upper.CL), width =0.2) +geom_vline(xintercept =1, linetype ="dashed", colour ="grey40") +labs(y ="Body Region", x ="Female / Male Odds Ratio", title ="Sex differences in carotenoid presence") +theme_bw() +theme(legend.position ="none")
3. Ordinal trait (TBD)
The ordinal trait is the percentage of body region with carotenoid presence. We will fit a beta-binomial and model to this trait, which is suitable for modeling proportions.
Now look at the difference in the proportion of carotenoid colour between females and males from the ordinal beta model (m5), and plot the model-based estimate with its 95% confidence interval on the response scale.
Code
emm_m5 <-emmeans(m5, ~ sex, type ="response")# pairwise contrast: Female vs Malem5_sex_OR <-contrast(emm_m5, method ="pairwise") |>summary(infer =TRUE, type ="response")# get summary with CIsex_OR <-as.data.frame(m5_sex_OR) |>rename(OR = odds.ratio, lower.CL = asymp.LCL, upper.CL = asymp.UCL)sex_OR
contrast OR SE df lower.CL upper.CL null z.ratio p.value
female / male 0.9169 0.0246 Inf 0.8699 0.9664 1 -3.235 0.0012
Confidence level used: 0.95
Intervals are back-transformed from the log odds ratio scale
Tests are performed on the log odds ratio scale
4.3 Case study 2: Evolution of plant hydraulic traits
We re-analysed the published study of Sanchez-Martinez et al. (2020) on the evolution of plant hydraulic traits using the phylogenetic generalized linear mixed models (PGLMMs) framework. The original study used a Bayesian MCMC approach to fit the models, but here we will use the glmmTMB packages to fit the models and assess different sampling distributions. NOTE: or just compare with MCMCglmm and show that glmmTMB is faster and has added functionality of gamma?
(log gaussian + gamma + MCMCglmm )
Data overview
Code
# Load plant hydraulic traits datasethydra.dat <-read_csv("data/HydraEvol2020.csv")# Have a look at distribution of traitshist(hydra.dat$Ks, breaks =50, main ="Distribution of Ks", xlab ="Ks")
Code
hist(hydra.dat$P50, breaks =50, main ="Distribution of P50", xlab ="P50")
Phylogenetic tree
Code
# Load bird tree (consensus tree = "combined tree")plant.tree <- ape::read.tree("data/genus-level_phylogeny.tre")# plot treeplotTree(plant.tree, ftype="i", fsize=0.4, lwd=1, type="fan")
Models
First let’s set up the models for glmmTMB: - Try gamma model - Log(response) gaussian
Brooks, M. E., Kristensen, K., van Benthem, K. J., Magnusson, A., Berg, C. W., Nielsen, A., Skaug, H. J.,Machler, M., & Bolker, B. M. (2017). glmmTMB balances speed and flexibility among packages for zero-inflated generalized linear mixed modeling. R Journal, 9 (2), 378–400. https://doi.org/10.32614/RJ-2017-066
Peter O. Dunn et al. ,Natural and sexual selection act on different axes of variation in avian plumage color.Sci. Adv.1,e1400155(2015).DOI:10.1126/sciadv.1400155
Hadfield, J. D. (2024, May). MCMCglmm: MCMC Generalised Linear Mixed Models. Retrieved October 7, 2024, from https://cran.r-project.org/web/packages/MCMCglmm/index.html
Montgomerie R. 2006. Analyzing colors. In Hill, G.E, and McGraw, K.J., eds. Bird Coloration. Volume 1 Mechanisms and measurements. Harvard University Press, Cambridge, Massachusetts.
Sanchez-Martinez, P., Martinez-Vilalta, J., Dexter, K. G., Segovia, R. A., & Mencuccini, M. (2020). Adaptation and coordinated evolution of plant hydraulic traits. Ecology Letters, 23 (11), 1599–1610. https://doi.org/10.1111/ele.13584
---title: "Supporting Information for: *Phylogenetic generalised linear mixed-effects modelling with glmmTMB R package*"author: "Coralie Williams, Maeve McGillycuddy, Szymek M. Drobniak, Ben M. Bolker, David I. Warton, Shinichi Nakagawa"date: "Last compiled on `r format(Sys.time(), '%d %B, %Y')`"format: html: page-layout: full toc: true toc-location: left toc-depth: 3 theme: materia embed-resources: true code-fold: show code-tools: true number-sections: true number-depth: 2 footnote-location: marginfavicon: "favicon.ico"editor_options: chunk_output_type: console---<!-- - make sure to use package name:: as much as possible --><!-- - make diagnostics of models using DHARMA (or glmmTMB diagnostics) --># Overview```{r setup}#| output: false#| warning: false#| label: packages#| code-overflow: wrap#| code-fold: trueknitr::opts_chunk$set( warning = FALSE, message = FALSE)pacman::p_load( rmarkdown, readr, dplyr, purrr, splitstackshape, tidyverse, tidyr, ggplot2, vroom, details, sessioninfo, devtools, readxl, forcats, stringr, ape, phytools, geiger, data.table, pavo, MASS, glmmTMB, brms, MCMCglmm, phyr, INLA, broom.mixed, knitr, bayestestR, performance, rbenchmark, here, DHARMa, emmeans)#setwd("C:/Users/z5394590/OneDrive - UNSW/Documents/Projects/phylo_glmm_sim/docs")options(digits = 4, scipen = 5)```This webpage provides supporting information for the manuscript "*Phylogenetic generalised linear mixed-effects modelling with glmmTMB R package*". The webpage is organised into three sections:(1) First, we describe the simulated models and data-generating mechanisms for each of the five evaluated R packages for fitting PGLMMs(2) We then use the simulated data to illustrate compilation and sampling runtimes for the Bayesian MCMC models.(3) Finally, we presents two case studies illustrating the application of these methods to ecological and evolutionary data, the first on bird color traits and the second on plant traits. For comments or questions, please contact the corresponding author at [coralie.williams@unsw.edu.au](mailto:coralie.williams@unsw.edu.au) or [coraliewilliams@outlook.com](coraliewilliams@outlook.com).# ModelsWe demonstrate and explain how to fit phylogenetic generalized mixed models (PGLMM) using five different packages which we used in our simulation study: `glmmTMB`, `phyr`, `MCMCglmm`, `brms`, and `INLA`. The models are fitted to one simulated dataset using a randomly generated phylogenetic tree and a set of model parameters. We compare the performance of these packages in terms of runtime, accuracy, and bias of the fixed effect mean and random effect variance estimates.<!-- ::: panel-tabset --><!-- ## Repeated measures per species -->We assume a set of trait measures $y_{ij}$ repeated measure (observation) $i$ and for each species $j$. The model is specified as follows:$$y_{ij} = \beta_0 + \beta_1 x_{ij} + n_j + p_j + \varepsilon_{ij}$$Where:- $y_{ij}$: trait response for measure (observation) $i$ in species $j$- $\beta_0, \beta_1$: fixed effects (intercept and slope)- $n_i \sim \mathcal{N}(0, \sigma^2_{\text{n}})$: species-level effect (non-phylogenetic)- $p_i \sim \mathcal{N}(0, \sigma^2_{\text{p}} \mathbf{A})$: species-level effect (phylogenetic), where $\mathbf{A}$ is the phylogenetic correlation matrix- $\varepsilon_{ij} \sim \mathcal{N}(0, \sigma^2_{e})$: residual error### Simulate dataFirst we specify the parameter values for one simulation run. We assume the following:```{r}### set up model parameters seed <-1b0 <-1# fixed effect interceptb1 <-1.5# fixed effect slopek.species <-30# number of speciesn.reps <-10# number of repeated measures per species (assuming a balanced design)sigma2.n <-0.25# variance of non-phylogenetic effectsigma2.p <-0.25# variance of phylogenetic effectsigma2.e <-0.2# residual error variance```Simulate data based on these parameter values:```{r}#| cache: true #| eval: true#| echo: trueset.seed(seed)# set up k.obs (number of observations per species)k.obs <- n.reps # assumes a balanced design# species id sp.id <-rep(seq_len(k.species), times=k.obs)# total number of observationsn <-length(sp.id)# simulate simple dataframe with covariate (x) variablex <-runif(n, 10, 20) dat <-data.frame(obs =1:n, x = x, species = sp.id)# simulate tree and obtain phylo matrixtree <-rtree(k.species, tip.label =seq_len(k.species))tree <-compute.brlen(tree, power=1) # power of 1 means ultra-metric treephylo.mat <-vcv(tree, corr =TRUE) ## we want a correlation matrix (bounded by -1 and 1)phylo.mat <- phylo.mat[order(as.numeric(rownames(phylo.mat))), order(as.numeric(rownames(phylo.mat)))]# Simulate response variable (phen) based on cofactor and phylogenetic matrixu.s <-rnorm(k.species, 0, sqrt(sigma2.n))[sp.id]u.p <-mvrnorm(1, mu=rep(0, k.species), Sigma=sigma2.p*phylo.mat)[sp.id]ei <-rnorm(n, 0, sqrt(sigma2.e))# get estimates of yyi <- b0 + b1*x + u.s + u.p + ei### append all to dataframedat <-cbind(dat, u.s, u.p, ei, b0, b1, yi)dat$phylo <- dat$species # phylo ID variable (same as species) - needs to be numeric to work with INLAdat$species <-factor(dat$species) # format species variable for modelsdat$sp <- dat$species # create sp variable (for phyr)dat$g <-1# add variable g constant (for glmmTMB)```View first five rows of the simulated dataset:```{r}head(dat)```### Run model with `glmmTMB`To fit a PGLMM using glmmTMB package we specify the species-level random effect with phylogenetic relatedness using the `propto` covariance structure. In the first part we specify the random intercept (`0 + species`), followed by the grouping variable `g` (which we assume here as constant), and then the phylogenetic correlation matrix `phylo.mat`.The model output will provide the fixed effect estimates, random effect variance estimates, and the residual variance estimate. The random effect variance estimates will be on the standard deviation scale, so we square them to get the variance estimates.```{r}#| cache: true #| eval: true#| echo: true#| label: Simulated glmmTMB model# repeated measures per speciestime.glmmTMB <-system.time({ model_glmmTMB <-glmmTMB(yi ~ x + (1|species) +propto(0+ species|g, phylo.mat),data = dat,REML =TRUE)})```Check if the model converged (i.e. returns `TRUE` if the Hessian is positive definite, for more details read here: https://cran.r-project.org/web/packages/glmmTMB/vignettes/troubleshooting.html):```{r}model_glmmTMB$sdr$pdHess```### Run model with `phyr`To fit the model using `phyr` package we use the `cov_ranef` argument to specify the phylogenetic tree directly. The random effect term `(1|sp__)` indicates that we want to include both phylogenetic and non-phylogenetic random effects. The underscores `__` indicate that we want to include phylogenetic correlations in the model.```{r}#| cache: true #| eval: true#| echo: false#| warnings: false#| label: Simulated phyr modeltime.phyr <-system.time({ model_phyr <-pglmm(yi ~ x + (1|sp__),cov_ranef =list(sp = tree),data = dat,REML =TRUE)})```### Run model with `MCMCglmm`To fit the MCMCglmm model we first we need to set up a precision matrix for the phylogenetic random effect which is required by MCMCglmm. We use the `inverseA` function from the `ape` package to obtain the inverse of the phylogenetic covariance matrix. The `prior` list specifies the priors for the random effects and residual variance. The `ginverse` argument is used to specify the inverse of the phylogenetic covariance matrix. Then we specify the number of iterations, burn-in, and thinning parameters for the MCMC sampling. Here we increased the number of iterations by 30 times the default value to ensure convergence and stability of the MCMC chains, but this could be further increased. Usually you would want to set these to some reasonably large values.```{r}#| cache: true #| eval: true#| echo: false#| label: Simulated MCMCglmm model# get precision phylo matrix and order rowsphylo.prec.mat <-inverseA(tree, nodes ="TIPS", scale =TRUE)$Ainvphylo.prec.mat <- phylo.prec.mat[order(as.numeric(rownames(phylo.prec.mat))),order(as.numeric(rownames(phylo.prec.mat)))]# set recommended priors with two random effectsprior <-list(G=list(G1=list(V=1,nu=1,alpha.mu=0,alpha.V=1000), G2=list(V=1,nu=1,alpha.mu=0,alpha.V=1000)),R=list(V=1,nu=0.02))# fit MCMCglmm modeltime.mcmc <-system.time({ model_mcmc <-MCMCglmm(yi ~ x,random =~species + phylo,family ="gaussian",ginverse =list(phylo = phylo.prec.mat),prior = prior,data = dat,verbose =FALSE,nitt =303000, # increase default by x30burnin =3000, # defaultthin =10) # default})```Next we check the effective sample size (ESS) and the Heidelberger diagnostic test which checks if the MCMC chains have converged and are stationary. The ESS should be above 400 for both fixed and random effects (Vehtari et al., 2021), and the Heidelberger diagnostic should return `TRUE` for all parameters.```{r}# check ESSmin(effective_sample(model_mcmc, effects ="fixed")$ESS)min(effective_sample(model_mcmc, effects ="random")$ESS)# check Heidelberger diagnosticfullchain <-cbind(as.mcmc(model_mcmc$Sol), as.mcmc(model_mcmc$VCV))heidel.diag(fullchain)```### Run model with `brms`To fit the model using the `brms` package we specify the random effects using the `(1|species)` term for non-phylogenetic random effects and `gr(phylo, cov = phylo.mat)` for phylogenetic random effects. The `data2` argument is used to pass the phylogenetic correlation matrix to the model (the same matrix format as for glmmTMB). We set the number of iterations, chains, and cores to reasonable values to ensure convergence and stability of the MCMC chains. Here we increased the number of iterations by 10 times the default value to ensure convergence and stability of the MCMC chains, but this could be further increased (and should be if ESS are low and Rhat values are higher than 1.01).```{r}#| cache: true #| eval: true#| echo: false#| warnings: false#| label: Simulated brms modeltime.brms <-system.time({ model_brms <-brm(yi ~ x + (1|species) + (1|gr(phylo, cov = phylo.mat)), #phylo.mat is the correlation matrixdata = dat,family =gaussian(),chains =4, # defaultiter =20000, # increased default x10cores =4, # equal to number of chainsdata2 =list(phylo.mat = phylo.mat))})```Check that the maximum effective sample size (ESS) of the model is high enough and check the Rhat value is below 1.01 (Vehtari et al. 2021):```{r}# check ESS min(bayestestR::effective_sample(model_brms, effects ="fixed")$ESS)min(bayestestR::effective_sample(model_brms, effects ="random")$ESS)# check RHat value is below 1.01 (Vehtari et al. 2021)max(rhat(model_brms))<1.01# alternatively, we can use the bayestestR package to check the diagnostics of fixed effectsprint(bayestestR::diagnostic_posterior(model_brms), digits =4)```### Run model with `INLA`For the `INLA` package we set up the model using the `f()` function to specify the random effects. The `model = "iid"` argument indicates that we want to use an independent and identically distributed (iid) random effect for species, while the `model = "generic0"` argument is used for the phylogenetic random effect. The `Cmatrix` argument is used to specify the phylogenetic correlation matrix. We also set up a penalizing complexity prior for the precision of the phylogenetic random effect (suggested by xxxx don't have a ref but can mention it?).```{r}#| cache: true #| eval: true#| echo: true#| label: Simulated INLA model# set up recommended penalizing complexity priors pcprior =list(prec =list(prior="pc.prec", param =c(20, 0.1)))time.inla <-system.time({ model_inla <-inla(yi ~ x +f(species, model ="iid") +f(phylo, ## this needs to be a numeric to workmodel ="generic0",Cmatrix = phylo.prec.mat,hyper=pcprior),family ="gaussian",data = dat)})fit_inla <-summary(model_inla)```### Extract model estimatesFirst we extract the covariate $x$ estimate and confidence interval for each model:```{r}#| cache: true#| eval: true#| echo: true#| label: Simulated model fixed effect estimates## Format run time for each model ---------time.phyr <-as.numeric(time.phyr[3])time.glmmTMB <-as.numeric(time.glmmTMB[3])time.brms <-as.numeric(time.brms[3])time.mcmc <-as.numeric(time.mcmc[3])time.inla <-as.numeric(time.inla[3])## Fixed effect estimates -------# phyrcoefs_phyr <-as.data.frame(fixef(model_phyr))coefs_phyr$conf.low[2] <- coefs_phyr$Value[2] - coefs_phyr$Std.Error[2]*1.96coefs_phyr$conf.high[2] <- coefs_phyr$Value[2] + coefs_phyr$Std.Error[2]*1.96# glmmTMBcoefs_tmb <-as.data.frame(confint(model_glmmTMB, parm="beta_"))# brmscoefs_brm <-as.data.frame(tidy(model_brms, effects="fixed", conf.int=TRUE))# MCMCglmm coefs_mcmc <-as.data.frame(tidy(model_mcmc, effects="fixed", conf.int=TRUE))# INLAcoefs_inla <-as.data.frame(fit_inla$fixed)## Combine fixed effect results -----------------res_fixed <-data.frame(model =c("phyr", "glmmTMB", "brms", "MCMCglmm", "INLA"),species_size = k.species,sample_size = n,run_time =c(time.phyr, time.glmmTMB, time.brms, time.mcmc, time.inla),b0 =rep(dat$b0[1], 5),b1 =rep(dat$b1[1], 5),mu =c(coefs_phyr$Value[2], coefs_tmb$Estimate[2], coefs_brm$estimate[2], coefs_mcmc$estimate[2], coefs_inla$mean[2]),mu_ci_low =c(coefs_phyr$conf.low[2], coefs_tmb$`2.5 %`[2], coefs_brm$conf.low[2], coefs_mcmc$conf.low[2], coefs_inla$`0.025quant`[2]),mu_ci_high =c(coefs_phyr$conf.high[2], coefs_tmb$`97.5 %`[2], coefs_brm$conf.high[2], coefs_mcmc$conf.high[2], coefs_inla$`0.975quant`[2]),stringsAsFactors =FALSE)## Show table of results of runtime and fixed effects ------kable(res_fixed, caption ="Runtime and fixed effect estimates of the simulated model",col.names =c("Model", "Species size", "Sample size", "Run time (s)", "b0", "b1", "Estimate (b1)", "CI low (b1)", "CI high (b1)"),digits =3,format ="html")```Extract the variance component estimates for each model and compare:```{r}#| cache: true#| eval: true#| echo: true#| label: Simulated model random effect estimates#### Random effect estimates -------# get phyr random effect variance estimates var_re_phyr <-c(as.numeric(model_phyr$ss[2])^2, #phylogenetic as.numeric(model_phyr$ss[1])^2, #non-phylogenetic as.numeric(model_phyr$ss[3])^2) #residual # combine into dataframesigma2_phyr <-data.frame(model ="phyr",group =c("phylo", "species", "Residual"),term ="var",estimate = var_re_phyr,std.error =NA,conf.low =NA, conf.high =NA)# get glmmTMB random effect variance estimates (by default it is on the standard deviation scale)re_tmb <-as.data.frame(confint(model_glmmTMB, parm="theta_"))species_tmb <- re_tmb[1, ]phylo_tmb <- re_tmb[2, ]# combine into dataframesigma2_tmb <-data.frame(model ="glmmTMB",group =c("phylo", "species", "Residual"),term ="var",estimate =c(phylo_tmb$Estimate^2, #phylo variance estimates species_tmb$Estimate^2, #non-phylo variance estimatessigma(model_glmmTMB)^2),std.error =NA, #conf.low =c(phylo_tmb$`2.5 %`, species_tmb$`2.5 %`, NA), # Replace with the residual var CI if availableconf.high =c(phylo_tmb$`97.5 %`, species_tmb$`97.5 %`, NA) # Replace with the residual var CI if available)# Compute variance, SE (delta method), and CI on variance scale)var_est <- re_tmb$Estimate^2var_se <-2* re_tmb$Estimate * (re_tmb$`97.5 %`- re_tmb$`2.5 %`) / (2*1.96)var_ci_low <- re_tmb$`2.5 %`^2var_ci_high <- re_tmb$`97.5 %`^2# Residual varianceresid_var <-sigma(model_glmmTMB)^2# Combine resultssigma2_tmb <-data.frame(model ="glmmTMB",group =c("phylo", "species", "Residual"),term ="var",estimate =c(var_est, resid_var),std.error =c(var_se, NA),conf.low =c(var_ci_low, NA),conf.high =c(var_ci_high, NA))# get brms random effect variance estimates (standard deviation scale)sigma_brms <-tidy(model_brms, effects="ran_pars")sigma2_brms <- sigma_brms %>%mutate(model="brms",term=str_replace(term, "sd", "var"),estimate=estimate^2) %>%##compute variance estimates dplyr::select(model, group, term, estimate, std.error, conf.low, conf.high)# get MCMCglmm random effect estimates (variance scale)sigma2_mcmc <-tidy(model_mcmc, effects="ran_pars", conf.int=TRUE)sigma2_mcmc <- sigma2_mcmc %>%mutate(model="MCMCglmm",group=str_replace(group,"animal", "phylo")) %>% dplyr::select(model, group, term, estimate, std.error, conf.low, conf.high)# get INLA random effect estimates (precision scale i.e. inverse variance)re_inla <-1/fit_inla$hyperparsigma2_inla <-data.frame(model ="INLA",group =c( "Residual", "species", "phylo"),term ="var",estimate = re_inla$mean,std.error = fit_inla$hyperpar$sd,conf.low = re_inla$`0.025quant`,conf.high = re_inla$`0.975quant`)# merge fixed results togethers2 <-as.data.frame(rbind(sigma2_phyr, sigma2_tmb, sigma2_brms, sigma2_mcmc, sigma2_inla))# get subsets for each groups2_phylo <- s2 %>%filter(group=="phylo")s2_sp <- s2 %>%filter(group=="species")s2_res <- s2 %>%filter(group=="Residual")## Combine random effect var estimates results -----------------res_rand <-data.frame(model =c("phyr", "glmmTMB", "brms", "MCMCglmm", "INLA"),species_size = k.species,sample_size = n,run_time =c(time.phyr, time.glmmTMB, time.brms, time.mcmc, time.inla),sigma2_phylo = s2_phylo$estimate,sigma2_species = s2_sp$estimate,sigma2_residual = s2_res$estimate,stringsAsFactors =FALSE)## Show table of results of runtime and random variance ------kable(res_rand, caption ="Runtime and random component variance estimates of the simulated model",col.names =c("Model", "Species size", "Sample size", "Run time (s)", "Phylo variance est.", "Non-phylo variance est.", "Residual variance est."),digits =3,format ="html")```# Bayesian MCMC model runtimeHere we provide a breakdown of the compilation and sampling runtimes for the Bayesian MCMC models using the above simulated repeated measures dataset. The runtimes are provided for each model fitted using the `MCMCglmm` and `brms` packages. Note: the compilation time is the time taken to compile the model, while the sampling time is the time taken to sample from the posterior distribution of the model parameters.First we set up the MCMCglmm model.### MCMCglmmSet up MCMCglmm tuning parameters - usually set to some reasonably large values through trial and error.```{r}#| eval: true#| echo: trueNITT <-100000BURNIN <-floor(0* NITT)THIN <-floor((NITT - BURNIN) /1500)```Run a model that prepares the run but does not start actual sampling to have the baseline timing of the model pre-run protocols.```{r}#| eval: true#| echo: trueruntime_1_pre <-benchmark("process"= { model_mcmcglmm_pre <-MCMCglmm(yi ~ x,random =~species + phylo,family ="gaussian",ginverse =list(phylo = phylo.prec.mat),prior = prior,data = dat,verbose =FALSE,nitt =1, burnin =0, thin =1 ) }, replications =1)runtime_1_pre$elapsed```Run full reasonable model (like above). Extract MCMC trace.```{r}#| cache: true #| eval: true#| echo: trueTHIN <-1model_mcmcglmm_max <-MCMCglmm(yi ~ x,random =~species + phylo,family ="gaussian",ginverse =list(phylo = phylo.prec.mat),prior = prior,data = dat,verbose =FALSE,nitt = NITT, burnin = BURNIN, thin = THIN)summary(model_mcmcglmm_max)par_mcmcglmm_max <- model_mcmcglmm_max$VCV[, "species"]plot(par_mcmcglmm_max)# geweke.plot(par_mcmcglmm_max)# geweke.diag(par_mcmcglmm_max)# heidel.diag(par_mcmcglmm_max)test <-raftery.diag(par_mcmcglmm_max)# update your run parametersNITT <- test$resmatrix[,"N"] + test$resmatrix[,"M"]THIN <-ceiling(test$resmatrix[,"N"] / test$resmatrix[,"Nmin"])BURNIN <- test$resmatrix[,"M"]runtime_1_run <-benchmark("process"= { model_mcmcglmm_1_run <-MCMCglmm(yi ~ x,random =~species + phylo,family ="gaussian",ginverse =list(phylo = phylo.prec.mat),prior = prior,data = dat,verbose =FALSE,nitt = NITT, burnin = BURNIN, thin = THIN ) }, replications =1)runtime_1_run$elapsed - runtime_1_pre$elapsed```### BRMS"Reasonable" `brms` parameters.```{r}NITTb <-10000BURNINb <-1000THINb <-1CHAINS <-1```Run a model that prepares the run but does not start actual sampling to have the baseline timing of the model pre-run protocols.```{r}#| cache: true #| warnings: falseruntime_2_pre <-benchmark("process"= { model_brms_1_pre <-brm(yi ~ x + (1|species) + (1|gr(phylo, cov = phylo.mat)),data = dat,data2 =list(phylo.mat = phylo.mat),chains = CHAINS,iter =2,warmup =1,thin =1 ) }, replications =1)runtime_2_pre$elapsed```Run "larger" model and convert posterior to MCMC object.```{r}#| cache: true #| warnings: falsemodel_brms_max <-brm(yi ~ x + (1|species) + (1|gr(phylo, cov = phylo.mat)),data = dat,data2 =list(phylo.mat = phylo.mat),chains = CHAINS,iter = NITTb,warmup = BURNINb,thin = THINb)summary(model_brms_max)par_brms_max <-as.mcmc(model_brms_max, pars ="sd_species__Intercept")[[1]]plot(par_brms_max)test <-raftery.diag(par_brms_max)# update your run parametersNITTb <- test$resmatrix[,"N"] + test$resmatrix[,"M"]THINb <-ceiling(test$resmatrix[,"N"] / test$resmatrix[,"Nmin"])BURNINb <- test$resmatrix[,"M"]runtime_2_run <-benchmark("process"= { model_brms_1_run <-brm(yi ~ x + (1|species) + (1|gr(phylo, cov = phylo.mat)),data = dat,data2 =list(phylo.mat = phylo.mat),chains = CHAINS,iter = NITTb,warmup = BURNINb,thin = THINb ) }, replications =1)runtime_2_run$elapsed - runtime_2_pre$elapsed```# Case studies## BackgroundThe following case studies illustrate how phylogenetic mixed models can be applied to diverse questions in ecology and evolution. The first examines patterns of plumage colour in birds, while the second explores macroevolutionary patterns in plant traits. Together, they demonstrate the flexibility of these methods across taxa and research questions.All models, results, and datasets presented in these case studies are **intended solely for illustrative purposes**. They are designed to demonstrate analytical approaches rather than to provide definitive biological insights. No substantive conclusions should be drawn from these analyses.## Case study 1: Bird color evolutionBirds are a valuable system for studying ecological and evolutionary processes because their diversity in form, behaviour, and coloration is well documented across many species and habitats. Variation in plumage colour, in particular, can provide insights into mechanisms such as sexual selection, camouflage, and signalling. By examining these traits, researchers can better understand how environmental and evolutionary pressures shape biodiversity.<!-- Maybe add some picture of birds: https://unsplash.com/s/photos/bird -->### Data overviewLoad data bird spectrum data:```{r}#| eval: true#| echo: true#| cache: true#| warnings: false#| label: Case study 1- Load data# Load bird spectrum data bird_data <-read_delim("data/Spec_IndivReg_Coralie.csv", delim =";", escape_double =FALSE, locale =locale(decimal_mark =","), trim_ws =TRUE)# summary of birds species and genuslength(table(bird_data$genus_original)) # 446 genuslength(table(bird_data$sci_name_Jetz)) # 949 species#length(table(bird_data$sci_name_original)) # 952 original species# Remove the specified species from bird_databird_data <- bird_data |>filter(!sci_name_Jetz %in%c("Basileuterus_rufifrons", "Malurus_lamberti", "Malurus_splendens"))########## Notes ########## missing values ---> not sure why?#na_count <- colSums(is.na(dat2))#na_count[na_count > 0]# individuals a-c are male and d-f are female#table(bird_data$sex, bird_data$individual)# number of measurements?#table(bird_data$Nmeasured)# duplicates per individual per body region#dup <- dat |># summarise(n = n(), # .by = c(wl, individual_nonrep, sex, body_region))#r <- dup[which(dup$n>1),]#table(r$individual_nonrep)```**Some notes:**- The dataset contains data for 949 bird species, with 446 unique genera.- Spectral values are normalised reflectance data.- `Nmeasured` is the number of measurements per body patch.- Each body region measured 5 times (then averaged).- These species seem to have two measurements per body region per individual, which we remove to facilitate the analysis.:``` Basileuterus_rufifrons, Malurus_lamberti, Malurus_splendens ```### Color data set-up#### Obtain wavelength datasetNow we want to create a wavelength dataset using the `pavo` package given the spectral reflectance data.```{r}#| eval: true#| echo: true#| cache: true#| label: Case study 1- Obtain wavelength dataset# set up wavelength dataset (wavelengths columns 300 to 474)dat <- bird_data |>pivot_longer(cols =`300`:`700`, names_to="wl", values_to="refl") |> dplyr::select(individual_nonrep, body_region, sex, wl, refl) |>mutate(wl =as.numeric(wl))# pivot so each column is an individual body region measurementdat2 <- dat |>pivot_wider(names_from =c("individual_nonrep", "sex", "body_region"),values_from ="refl", names_sep =".")# create spectral dataset with pavospecs <-as.rspec(dat2)# some examples of individual birds body region color spectrumplot(Acrocephalus_palustris_a.Male.throat ~ wl, type="l", data=specs, title="Acrocephalus palustris (throat)")plot(Alle_alle_a.Male.throat ~ wl, type="l", data=specs, title="Alle alle (throat)")plot(Aplonis_metallica_a.Male.wing_cov ~ wl, type ="l", data = specs, title="Aplonis metallica (wing covert)")# use procspec to adjust negative values by shifting them by 10# (this maybe due to darker colors)specs <-procspec(specs, fixneg="addmin")plot(specs, select =10)```#### Obtain spectral shape descriptorsUsing the wavelength dataset we can now obtain spectral shape descriptors. These descriptors will be used in the model. Here we will focus on four descriptors of interest: brightness (B1), spectral slope (S1), spectral curvature (S9), and hue (H4). More information about the spectral shape descriptors can be found here: https://book.colrverse.com/spectral-shape-descriptors.html```{r}#| eval: true# Obtain spectral shape descriptorsspec.des <-summary(specs, subset =c("B1", "B2", "S9", "H4")) ## using subset makes it faster to run by selecting the shapes of interestdev.off()# distribution of B1ggplot(spec.des, aes(x = B1)) +geom_histogram(bins =30, colour ="black", fill ="grey") +labs(x ="B1", title ="B1: Total brightness") +theme_bw()# distribution of S9ggplot(spec.des, aes(x = S9)) +geom_histogram(bins =30, colour ="black", fill ="grey") +labs(x ="S9", title ="S9: Carotenoid chroma") +theme_bw()# vdistribution o fH4ggplot(spec.des, aes(x = H4)) +geom_histogram(bins =30, colour ="black", fill ="grey") +labs(x ="H4", title ="H4: Hue (segment classification)") +theme_bw()```The first dataset is the shape descriptors dataset, which we will use to model the continuous trait (brightness). ```{r}#| eval: true#| echo: truespec.des$rowname <-rownames(spec.des)spec.dat <- spec.des |>mutate(sex =case_when(grepl("Male", rowname) ~"male",grepl("Female", rowname) ~"female"),species =gsub("\\_[a-f]\\..*$", "", rowname),body_region =str_extract(rowname, "[a-z]+$") ) write.csv(spec.dat, file="data/spec_data_for_model.csv")```#### Obtain carotenoid datasets: binary and ordinal traitsWe will use the carotenoid dataset to model the presence/absence of carotenoid coloration in birds. This dataset is based on the spectral reflectance data and the spectral shape descriptors.Let's set up the dataset for modelling a binary trait (absence/presence of carotenoid color):```{r}#| cache: true# 1) get human visual model (CIE 10 degree observed under D65 "day light")vm <-vismodel(specs, visual ="cie10", illum ="D65", bkg ="ideal", relative =FALSE)# 2) convert to CIELAB colorspace (to get hue and chroma)lab <-colspace(vm, space ="cielab") # 3) get chroma and hue angle (converting CIELAB to CIELCh)L <- lab$La <- lab$ab <- lab$bC <-sqrt(a^2+ b^2)h <- (atan2(b, a) *180/ pi) %%360# degrees in 0 to 360# 4) thresholds for carotenoid range and saturationh_lower <-330# start of redh_upper <-100# end of yellowC_min <-15# min chromaL_min <-20# avoid very dark samples# 5) Hue range test in_range <- h >= h_lower | h <= h_upper# 6) COmpute binary trait: 1 = carotenoid like colour present, 0 = absentpresent <-as.integer(in_range & C >= C_min & L >= L_min)# set up datasetcarot.dat.all <-data.frame(rowname =rownames(lab),L = L, a = a, b = b, C = C, h = h,carotenoid = present)# merge with spec.data based on "rowname" columncarot.dat.all <-merge(spec.dat, carot.dat.all, by.x ="rowname", by.y ="rowname", all =TRUE)# create individual id variablecarot.dat.all$indiv_rep <-sub(".*_([a-f])\\..*$", "\\1", carot.dat.all$rowname)# number of observations with carotenoid like colourtable(carot.dat.all$carotenoid)# # checks# carot.dat.all |># filter(L>20 & C>12 & in_range) |># arrange(desc(L))# # check bird cardinalis cardinalis (should be yellow or red?)# carot.dat.all |> # filter(grepl("Cardinalis_cardinalis", rowname)) |> # arrange(desc(h))# Plotggplot(carot.dat.all, aes(h, C)) +geom_point(data =subset(carot.dat.all, carotenoid ==0),shape =21, size =1.5, fill ="grey85", colour ="grey70", alpha =0.6) +geom_point(data =subset(carot.dat.all, carotenoid ==1),aes(fill = L, colour = L), shape =21, size =1.8, stroke =0.4, alpha =0.95) +scale_fill_gradient(name ="L*", limits =c(15, 50), low ="darkred", high ="lightcoral") +scale_colour_gradient(guide ="none", limits =c(15, 50), low ="darkred", high ="lightcoral") +labs(x ="Hue (degrees)", y ="Chroma (C*)", title ="Carotenoid presence") +theme_bw()```Let's save the dataset for modelling later on:```{r}carot.dat <- carot.dat.all# save as csvwrite.csv(carot.dat, file="data/carotenoid_data_for_model.csv", row.names =FALSE)```Now let's obtain the dataset where we summarise for each individual bird the proportion of body region with carotenoid color presence (ordinal data trait), and save it for modelling later on:```{r}carot.ordinal <- carot.dat.all |>group_by(species, sex, indiv_rep) |>summarise(prop_carotenoid =mean(carotenoid), .groups ="drop")# save as csvwrite.csv(carot.ordinal, file="data/ordinal_data_for_model.csv", row.names =FALSE)```### Phylogenetic correlation matrix set-upLoad and view phylogenetic tree:```{r}#| output: true#| warnings: true#| eval: true#| label: Case study 1- Phylogenetic tree#| # Load bird tree (consensus tree = "combined tree")bird.tree <-read.tree("data/Stage2_Hackett_MCC_no_neg.tre")### Prune bird treebird.pruned <-keep.tip(bird.tree, bird_data$sci_name_Jetz)# check whether names match in data and treecheck <-name.check(bird.pruned, bird_data$sci_name_Jetz, sort(bird.pruned$tip.label))# plot treeplotTree(bird.pruned, ftype="i", fsize=0.4, lwd=1, type="fan")dev.off()```Set up correlation matrix for glmmTMB model and check it corresponds to the species labels in the data:```{r}#| eval: true#| echo: true# set up phylogenetic correlation matrixphylo.mat <-vcv(bird.pruned, corr =TRUE) phylo.mat <- phylo.mat[sort(rownames(phylo.mat)), sort(rownames(phylo.mat))]saveRDS(phylo.mat, file ="data/phylo_matrix.rds")# checks # length(colnames(phylo.mat))==length(table(spec.dat$species))# all(head(rownames(phylo.mat))==head(colnames(phylo.mat)))# head(table(spec.dat$species))```### Models(1) Continuous trait (brightness): we compare models with distributions of gaussian, lognormal, gamma, and skew normal.(2) Binary trait (absence/presence of color): binomial distribution.(3) Ordinal trait (100% of body region with color): beta binomial.#### 1. Continuous trait```{r}#| eval: true# load dataspec.dat <-read_csv("data/spec_data_for_model.csv")phylo.mat <-readRDS("data/phylo_matrix.rds")# load library library(glmmTMB)# add grouping variable (set it to 1) - this is necessary to fit the glmmTMB modelspec.dat$g <-1```##### Modelling total brightness (B1)The brightness trait is right-skewed (as shown above), which is consistent with multiplicative evolutionary change. To identify an appropriate sampling distribution, we will fit four models with an identical linear predictor and random effects structure: 1. Gaussian to model $\log(B1)$ 2. Gamma with a log link to model $B1$ For each model we will examine simulated standardised residuals using \texttt{DHARMa} to assess nonlinearity and heteroscedasticity. If two or more models perform similarly, we will prefer the model with clearer interpretation on the original scale.```{r}#| output: true#| warnings: true#| eval: true#| cache: true # # Little checks ---------# library(fitdistrplus)# descdist(spec.dat$B1, discrete=FALSE)# gamma_dist <- fitdist(spec.dat$B1, "gamma")# plot(gamma_dist)# norm_dist <- fitdist(log(spec.dat$B1), "norm")# plot(norm_dist)# lnorm_dist <- fitdist(spec.dat$B1, "lnorm")# plot(lnorm_dist)# Fit models --------------# normaltime_norm <-system.time( m1 <-glmmTMB(log(B1) ~ body_region * sex + (1|species) +propto(0+ species|g,phylo.mat),family =gaussian(),data = spec.dat) # don't use REML (to get AIC) )# Gamma distributiontime_gamma <-system.time( m2 <-glmmTMB(B1 ~ body_region * sex + (1|species) +propto(0+ species|g,phylo.mat),family =Gamma(link ="log"),data = spec.dat))# # Lognormal ---> not working get an error that the link function is not supported?# t_lnorm <- system.time(# modb1_lnorm <- glmmTMB(B1 ~ body_region * sex + (1|species) + propto(0 + species|g,phylo.mat), family = lognormal(link = "log"), data = spec.dat)# )# Get model info ---------------------------# check whether model has postive definite hessianb1_output <-data.frame(model =c("Gaussian (log)", "Gamma"),convergence =c(m1$sdr$pdHess, m2$sdr$pdHess),runtime =c(time_norm[["elapsed"]], time_gamma[["elapsed"]]))b1_output```##### Model diagnosticsLet's check residual plots with the DHARMa packages. First, the log(B1) assuming normal distribution residual checks:```{r}#| cache: true res_gauss <- DHARMa::simulateResiduals(fittedModel = m1)plot(res_gauss)```The model checks assuming Gamma distribution and log link function:```{r}#| cache: true res_gamma <- DHARMa::simulateResiduals(fittedModel = m2)plot(res_gamma)```We found that the Gamma distribution shows improve model fit as the residual plots ##### Model estimatesLet's obtain the estimate of the phylogenetic signal in the :```{r}#| output: true#| cache: true m2_re <-as.data.frame(confint(m2, parm="theta_"))sigma2_s <- m2_re$Estimate[1] # non-phylogenetic variancesigma2_p <- m2_re$Estimate[2] # phylogenetic variance estimatep.signal <- sigma2_p / (sigma2_p + sigma2_s)p.signal```On the log mean scale of the Gamma model, the phylogenetic species effect explained 69.8% of the total species level random effect variance i.e. this represents the degree of phylogenetic signal in the overall variance sourced from species.To assess differences in total brightness between sexes for each body region, we compute marginal means from the fitted model and perform pairwise comparisons between females and males. The resulting ratios (female/male) and their confidence intervals are then plotted to visualise the magnitude and direction of differences across body regions.```{r}#| output: true#| warnings: true#| eval: trueemm_b1 <-emmeans(m2, ~ sex | body_region, type ="response")b1_sex_diff <-contrast(emm_b1, method ="pairwise")## Transform to response scale as a ratiosex_diff_df <-as.data.frame(b1_sex_diff)sex_diff_df$lower.CL <- sex_diff_df$ratio - sex_diff_df$SEsex_diff_df$upper.CL <- sex_diff_df$ratio + sex_diff_df$SE## plot pairwise differences between female and male total brightnesssex_diff_df |>arrange(ratio) |>mutate(body_region =factor(body_region, unique(body_region), ordered = T)) |>ggplot(aes(y = body_region, x = ratio, color = body_region)) +geom_point(size =3) +geom_errorbar(aes(xmin = lower.CL, xmax = upper.CL), width =0.2) +geom_vline(xintercept =1, linetype ="dashed", colour ="grey40") +labs(y ="Body Region", x ="Female / Male Ratio", title ="Brightness") +theme_bw() +theme(legend.position ="none") # Optional: remove legend if redundant```Ratios $>1$ indicate greater brightness in females, while ratios $<1$ indicate greater brightness in males.#### 2. Binary traitBinary pigment traits, such as the presence of carotenoid colors, may be associated with distinct evolutionary and ecological drivers. Here we want to model the absence or presence of carotenoid across all body regions accounting for sex and including species- and phylogeny-level random effects. We will use the carotenoid dataset to model based on the spectral reflectance data which we derived above.##### Modelling carotenoid presenceWe will fit two models:- Binomial model with a logit link function- beta binomial model with a logit link function```{r}#| eval: true#| echo: true#| cache: true# Fit models --------------# load datcarot.dat <-read_csv("data/carotenoid_data_for_model.csv")carot.dat$g <-1# binomial modeltime_binom <-system.time( m3 <-glmmTMB(carotenoid ~ body_region * sex + (1|species) +propto(0+ species|g,phylo.mat),family =binomial(link ="logit"),data = carot.dat))# beta binomialtime_bbinom <-system.time( m4 <-glmmTMB(carotenoid ~ body_region * sex + (1|species) +propto(0+ species|g,phylo.mat),family =betabinomial(link ="logit"),data = carot.dat))# Get model info -----------carotmod_output <-data.frame(model =c("Binomial", "Beta binomial"),convergence =c(m3$sdr$pdHess, m4$sdr$pdHess),runtime =c(time_binom[["elapsed"]], time_bbinom[["elapsed"]]),AIC =c(AIC(m3), AIC(m4)))carotmod_output ```##### Model diagnosticsLet's check residual plots with the DHARMa packages. First, let's get the residual plots for the binomial model:```{r}#| cache: true res_binom <- DHARMa::simulateResiduals(fittedModel = m3)plot(res_binom)```Now the model checks assuming zero inflated binomial distribution and log link function:```{r}#| cache: true res_bbinom <- DHARMa::simulateResiduals(fittedModel = m4)plot(res_bbinom)```We found that the binomial model has improved model fit in AIC and in the residual plots.##### Model estimatesGet model estimates:```{r}library(emmeans)# Estimated marginal means on the response scale (odds)emm_m3 <-emmeans(m3, ~ sex | body_region, type ="response")# get contrasts (odds ratios)m3_sex_diff <-contrast(emm_m3, method ="pairwise") |>summary(infer =TRUE, type ="response")# Put into data frame for plottingsex_diff_df <-as.data.frame(m3_sex_diff) |>rename(ratio = odds.ratio, lower.CL = asymp.LCL, upper.CL = asymp.UCL)# Plotsex_diff_df |>arrange(ratio) |>mutate(body_region =factor(body_region, unique(body_region), ordered =TRUE)) |>ggplot(aes(y = body_region, x = ratio, colour = body_region)) +geom_point(size =3) +geom_errorbar(aes(xmin = lower.CL, xmax = upper.CL), width =0.2) +geom_vline(xintercept =1, linetype ="dashed", colour ="grey40") +labs(y ="Body Region", x ="Female / Male Odds Ratio", title ="Sex differences in carotenoid presence") +theme_bw() +theme(legend.position ="none")```#### 3. Ordinal trait (TBD)The ordinal trait is the percentage of body region with carotenoid presence. We will fit a beta-binomial and model to this trait, which is suitable for modeling proportions.##### Modelling carotenoid proportion per body region```{r}#| cache: true # load dataord.dat <-read_csv("data/ordinal_data_for_model.csv")ord.dat$g <-1# add grouping variable # fit model time_ord <-system.time( m5 <-glmmTMB(prop_carotenoid ~ sex + (1|species) +propto(0+ species|g,phylo.mat),family =ordbeta(),data = ord.dat))# outputdata.frame(model ="Ordinal beta",convergence = m5$sdr$pdHess,runtime = time_ord[["elapsed"]])```##### Model diagnosticsLook at residual diagnostic plots:```{r}#| cache: true res_ord <- DHARMa::simulateResiduals(fittedModel = m5)plot(res_ord)```##### Model estimatesNow look at the difference in the proportion of carotenoid colour between females and males from the ordinal beta model (m5), and plot the model-based estimate with its 95% confidence interval on the response scale.```{r}emm_m5 <-emmeans(m5, ~ sex, type ="response")# pairwise contrast: Female vs Malem5_sex_OR <-contrast(emm_m5, method ="pairwise") |>summary(infer =TRUE, type ="response")# get summary with CIsex_OR <-as.data.frame(m5_sex_OR) |>rename(OR = odds.ratio, lower.CL = asymp.LCL, upper.CL = asymp.UCL)sex_OR```## Case study 2: Evolution of plant hydraulic traitsWe re-analysed the published study of Sanchez-Martinez et al. (2020) on the evolution of plant hydraulic traits using the phylogenetic generalized linear mixed models (PGLMMs) framework. The original study used a Bayesian MCMC approach to fit the models, but here we will use the `glmmTMB` packages to fit the models and assess different sampling distributions. NOTE: or just compare with MCMCglmm and show that glmmTMB is faster and has added functionality of gamma?(log gaussian + gamma + MCMCglmm )### Data overview```{r}# Load plant hydraulic traits datasethydra.dat <-read_csv("data/HydraEvol2020.csv")# Have a look at distribution of traitshist(hydra.dat$Ks, breaks =50, main ="Distribution of Ks", xlab ="Ks")hist(hydra.dat$P50, breaks =50, main ="Distribution of P50", xlab ="P50")```### Phylogenetic tree```{r}# Load bird tree (consensus tree = "combined tree")plant.tree <- ape::read.tree("data/genus-level_phylogeny.tre")# plot treeplotTree(plant.tree, ftype="i", fsize=0.4, lwd=1, type="fan")```### ModelsFirst let's set up the models for glmmTMB:- Try gamma model- Log(response) gaussian```{r}# log_Ks ~ animal + genus.Rand# log_negP50 ~ animal + genus.Rand# log_Hv ~ animal + genus.Rand# log_negMinWP_md ~ animal + genus.Rand```Same models with MCMCglmm```{r}```Compare model outputs + run time### Model results```{r}```# ReferencesBrooks, M. E., Kristensen, K., van Benthem, K. J., Magnusson, A., Berg, C. W., Nielsen, A., Skaug, H. J.,Machler, M., & Bolker, B. M. (2017). glmmTMB balances speed and flexibility among packages for zero-inflated generalized linear mixed modeling. R Journal, 9 (2), 378–400. https://doi.org/10.32614/RJ-2017-066Peter O. Dunn et al. ,Natural and sexual selection act on different axes of variation in avian plumage color.Sci. Adv.1,e1400155(2015).DOI:10.1126/sciadv.1400155Hadfield, J. D. (2024, May). MCMCglmm: MCMC Generalised Linear Mixed Models. Retrieved October 7, 2024, from https://cran.r-project.org/web/packages/MCMCglmm/index.htmlMontgomerie R. 2006. Analyzing colors. In Hill, G.E, and McGraw, K.J., eds. Bird Coloration. Volume 1 Mechanisms and measurements. Harvard University Press, Cambridge, Massachusetts.Sanchez-Martinez, P., Martinez-Vilalta, J., Dexter, K. G., Segovia, R. A., & Mencuccini, M. (2020). Adaptation and coordinated evolution of plant hydraulic traits. Ecology Letters, 23 (11), 1599–1610. https://doi.org/10.1111/ele.13584# Session information```{r}#| label: Reproducibility-SessionInfo-R-environment#| fig-align: "center"#| out-width: '100%'#| results: asis#| message: false#| warnings: falselibrary(sessioninfo)library(details)si <-session_info()si$packages <- si$packages # |> filter(package %in% c("metafor", "ape", "clubSandwich", "Matrix", "corpcor", "dplyr", "kableExtra", "xtable", "rotl", "Hmisc", "lattice"))details(si, summary ='Current session info', open =FALSE)```
